Evaluate the line integral z c z2 dx 2y dy 2xz dz where c is any closed curve in the space.

Some other information should be given, I have added those necessary information, should in case you come across a similar question on line integral.

Step-by-step explanation:

To evaluate line integral, some other information has to be given, Assume (so when you see a similar question, you will know how to evaluate line integral) as part of the question X =t2, Y = 2t and Z = 3t and at interval 0≤t≤1

A step by step explanation and calculation has been attached.

(b) 240

step-by-step explanation:

first, we use the pythagorean formula using the given side lengths to find x. then we use x to find the side lengths. then we add the side lengths to find the perimeter.

pythagorean theorem formula:

a^2 + b^2 = c^2

(x - 20)^2 + (x - 40)^2 = x^2

x^2 - 40x + 400 + x^2 - 80x + 1600 = x^2

x^2 -120x + 2000 = 0

(x - 100)(x - 20) = 0

x - 100 = 0 or x - 20 = 0

x = 100 or x = 20

we see that the solution x = 20 must be discarded because it will give a negative side length and a side length of 0. the only valid solution is x = 100.

perimeter = sum of the lengths of the sides

perimeter = x + x - 20 + x - 40

perimeter = 3x - 60

replace x with 100.

perimeter = 3(100) - 60

perimeter = 300 - 60

perimeter = 240

the answer is b